Infinity

Infinity
Imagine infitinite possibilities if infinity existed

I will not dwindle on the mathematical terms of infinity and its strict definitions. However, there are couple of (technical) properties of infinity that I would like to mention before I proceed: (1) Infinity is very different from everything we know in everyday life and is not governed by the same rules which most of us are familiar with, (2) there are different types of infinities. If not interested in any of these technical stuff, you can skip the following two paragraphs.

(1) Most people think infinity as equivalent to some very very large number and equates its magnitude with the vast size of the universe. But that is misleading, even though some of the approximations are good enough (or sometimes almost exact) for practical reasons. However, there is not a single number that can represent infinity. Let's take 100 billion, large enough number for non-scientific people, but it is equal to only 1/500 of the number of cells in our body (estimated at anything between 10 trillion to 100 trillion and I have taken 50 trillion for my calculation). So let's take 50 trillion, but that is only 1/140000000000000 (that's 14 followed by 13 zeros) of the number of atoms in our body (estimated value for 70kg human body). So on, we can always dwarf any chosen number and not just find a bigger number.
Some people talk of the infinite space as the universe. In earlier times, our understanding of the universe consisted of only Milky Way galaxy, which is only one of the billions of galaxies in our universe. Even the entire universe is only 13.5 billion years old (so the largest possible size of the universe can be put at 27 billion light-years across). So again, it is not large enough to represent the infinite size. It is true, that there are people who believe in multiverse (i.e. many universes), but we will (possibly) never know whether that's the reality or not and we will need real infinite number of universes combined together to represent the size of the infinite space. But at this moment, there is no evidence for that reality.


(2) There are different types of infinites. There are infinite even numbers. But even numbers' cardinality is no less than the cardinality of integers, even though set of even numbers is a subset of set of integers. That's because we can establish 1-1 correspondance between the members of 2 sets: for example. 1 to 2, 2 to 4, 3 to 6... that's a 1-1 correspondance from the set of integers to the set of even numbers. So, in a sense, two sets are equal. However, the cardinality of set of numbers between 0 and 1 (including the irrational numbers) is greater than the cardinality of set of integers, meaning we cannot establish a 1-1 correspandance between the members of the two sets.


Now, let me get to the main points: what is possible if infinity was real? I consider two cases: one with infinite number of people living (and of course, that's only possible with infinite amount of resources available) and another one with someone with an infinite amount of resources.

First, if there were infinite number of people living, then any number of people can be rich through Ponzi-scheme and Pyramid-scheme (note, in some places they don't distinguish between the two). Both the schemes rely on that there will be some more people to pay for your divident/reward/payback after you join the scheme as both schemes use new-comers' money to pay for already-members of the scheme. So the schemes sustain as long as there is a pool of people who can join the scheme and that's not possible in real-life, which is limited by the total population in the world. However, both schemes can sustain in a world where infinite number of people live and therefore any number of people can be rich as there will always be infinite number of people not in the scheme. Unfortunately, there will always be people outside the scheme, which means there will always be poor people and the process will need to last for infinitely long time. But, realistically such a world doesn't exist.

Second, imagine someone with an infinite amount of resources. Then, such a person (let's call him Richard) can always win any kind of betting-game, even with very low probabilty of winning. Let's imagine a game with the probability of 1/100 to double your money. It is very unlikely that anyone will play such a game, as they are going to think that's such a rip-off. But our Richard can beat the system and can still make money, not because of his luck, but because of his infinite endowment. He bets £1. If he wins, then it's over. But if he loses, he bets £2 next time. If he wins this time, he gets £4 back, making a profit of £1 (£4 minus the £2 bet minus the initial loss of £1). But if he loses, he bets £4 next time... and so on, doubling his bet till he wins and he is guaranteed to win if he waits long enough and that is only possible provided he has an infinite amount of money available to him. But then, such a person would not be that desperate to make only £1.

Let me finish my little article by introducing quite well known Hilbert's Hotel, consider a hotel with infinite number of rooms and every room is occupied. Now, assume that every guest brings a friend and would like to get a room for their friends. That is not possible for a hotel fully occupied in real life. But it is not a problem with Hilbert's hypothetical hotel. You move the person in Room 1 to Room 2, and the person in Room 2 to Room 4, etc... Now all the guests are in even numbered rooms and, therefore odd numbered rooms are empty. So their friends can get a room. This is an example that infinity is NOT governed by the same rules as numbers (apple+apple=2 apples, but infinity+infinity≠2infinities).